Problem: Find the distance between the point ${(-3, -2)}$ and the line $\enspace {y = -\dfrac{1}{2}x - 6}\thinspace$. {1} {2} {3} {4} {5} {6} {7} {8} {9} {\llap{-}2} {\llap{-}3} {\llap{-}4} {\llap{-}5} {\llap{-}6} {\llap{-}7} {\llap{-}8} {\llap{-}9} {1} {2} {3} {4} {5} {6} {7} {8} {9} {\llap{-}2} {\llap{-}3} {\llap{-}4} {\llap{-}5} {\llap{-}6} {\llap{-}7} {\llap{-}8} {\llap{-}9}
Answer: First, find the equation of the perpendicular line that passes through ${(-3, -2)}$ The slope of the blue line is ${-\dfrac{1}{2}}$ , and its negative reciprocal is ${2}$ Thus, the equation of our perpendicular line will be of the form $\enspace {y = 2x + b}\thinspace$ We can plug our point, ${(-3, -2)}$ , into this equation to solve for ${b}$ , the y-intercept. $-2 = {2}(-3) + {b}$ $-2 = -6 + {b}$ $-2 + 6 = {b} = 4$ The equation of the perpendicular line is $\enspace {y = 2x + 4}\thinspace$ We can see from the graph (or by setting the equations equal to one another) that the two lines intersect at the point ${(-4, -4)}$ . Thus, the distance we're looking for is the distance between the two red points. The distance formula tells us that the distance between two points is equal to: $\sqrt{( x_{1} - x_{2} )^2 + ( y_{1} - y_{2} )^2}$ Plugging in our points ${(-3, -2)}$ and ${(-4, -4)}$ gives us: $\sqrt{( {-3} - {-4} )^2 + ( {-2} - {-4} )^2}$ $= \sqrt{( 1 )^2 + ( 2 )^2} = \sqrt{5} $ The distance between the point ${(-3, -2)}$ and the line $\thinspace {y = -\dfrac{1}{2}x - 6}\enspace$ is $\thinspace\sqrt{5}$.